Let $X$ be an abelian variety of dimension $g$ over a field $k$. In general, the group $\textrm{Aut}_k(X)$ of automorphisms of $X$ over $k$ is not finite. But if we fix a polarization $\mathcal{L}$ on $X$, then the group $\textrm{Aut}_k(X,\mathcal{L})$ of automorphisms of the polarized abelian variety $(X,\mathcal{L})$ over $k$ is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over some field $k.$ In one of earlier works, a complete classification of such finite groups was given for the case when $k$ is a finite field, $g$ is an odd prime, and $X$ is simple. One interesting thing is that the maximal such a finite group was a cyclic group of order $4g+2$ only when $g$ is a Sophie Germain prime. Another notable thing is the fact that the abelian variety that was constructed to achieve the maximal cyclic group splits over $\overline{k}$, an algebraic closure of $k.$ \\
In this talk, we provide a construction of an absolutely simple abelian variety of dimension $g$ ($g$ being a Sophie Germain prime) over a finite field $k$, which attains the maximal automorphism group. This can be regarded as the counterpart for the previous construction. Also, we briefly describe the asymptotic behavior of the characteristic of the base field $k$ for which we can give such a construction. Finally, if time permits, we take a closer look at the case of $g=5$ by introducing the Newton polygon of an abelian variety of dimension $5.$
(If you want to join the seminar, please contact Bo-Hae Im to get the zoom link.)
|
